Reinforcement Learning. Reinforcement learning and HMMs. Hidden Markov Models (HMMs) are appropriate when our agent models the world as follows

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1 Reinforcement Learning Reinforcement learning and HMMs We now examine: some potential shortcomings of hidden Markov models, and of supervised learning; an extension know as the Markov Decision Process (MDP); the way in which we might learn from rewards gained as a result of acting within an environment; specific, simple algorithms for performing such learning, and their convergence properties. Reading: Russell and Norvig, chapter 21. Mitchell chapter 13. Copyright c Sean Holden Hidden Markov Models (HMMs) are appropriate when our agent models the world as follows Pr(S0) S0 S1 S2 S3 E1 Pr(St St 1) Pr(Et St) and only wants to infer information about the state of the world on the basis of observing the available evidence. This might be criticised as un-necessarily restricted, although it is very effective for the right kind of problem. E2 E3 Reinforcement learning and supervised learning Reinforcement learning: the basic case Supervised learners learn from specifically labelled chunks of information: We now begin to model the world in a more realistic way as follows: x??? S0 S1 S2 S3 (x1, 1) (x2, 1) (x3, 0). In any state: Perform an action a to move to a new state. (There may be many possibilities.) Receive a reward r depending on the start state and action. The agent can perform actions in order to change the world s state. This might also be criticised as un-necessarily restricted: there are other ways to learn. If the agent performs an action in a particular state, then it gains a corresponding reward.

2 Deterministic Markov Decision Processes Formally, we have a set of states S = {s 1, s 2,...,s n } and in each state we can perform one of a set of actions We also have a function A = {a 1, a 2,...,a m }. S : S A S such that S(s, a) is the new state resulting from performing action a in state s, and a function R : S A R such that R(s, a) is the reward obtained by executing action a in state s. Deterministic Markov Decision Processes From the point of view of the agent, there is a matter of considerable importance: The agent does not have access to the functions S and R. It therefore has to learn a policy, which is a function p : S A such that p(s) provides the action a that should be executed in state s. What might the agent use as its criterion for learning a policy? Measuring the quality of a policy Say we start in a state at time t, denoted s t, and we follow a policy p. At each future step in time we get a reward. Denote the rewards r t, r t+1,... and so on. A common measure of the quality of a policy p is the discounted cumulative reward V p (s t ) = ǫ i r t+i = r t + ǫr t+1 + ǫ 2 r t+2 + where 0 ǫ 1 is a constant, which defines a trade-off for how much we value immediate rewards against future rewards. Measuring the quality of a policy Other common measures are the average reward 1 T lim r t+i T T and the finite horizon reward T In these notes we will only address the discounted cumulative reward. r t+i The intuition for this measure is that, on the whole, we should like our agent to prefer rewards gained quickly.

3 Two important issues Note that in this kind of problem we need to address two particularly relevant issues: The temporal credit assignment problem: that is, how do we decide which specific actions are important in obtaining a reward? The exploration/exploitation problem. How do we decide between exploiting the knowledge we already have, and exploring the environment in order to possibly obtain new (and more useful) knowledge? The optimal policy Ultimately, our learner s aim is to learn the optimal policy p opt = argmaxv p (s) p for all s. We will denote the optimal discounted cumulative reward as V opt (s) = V p opt (s). How might we go about learning the optimal policy? We will see later how to deal with these. Learning the optimal policy The only information we have during learning is the individual rewards obtained from the environment. We could try to learn V opt (s) directly, so that states can be compared: Consider s as better than s if V opt (s) > V opt (s ). However we actually want to compare actions, not states. Learning V opt (s) might help as p opt (s) = argmax [R(s, a) + ǫv opt (S(s, a))] a but only if we know S and R. The Q function The trick is to define the following function: Q(s, a) = R(s, a) + ǫv opt (S(s, a)) This function specifies the discounted cumulative reward obtained if you do action a in state s and then follow the optimal policy. As p opt (s) = argmax Q(s, a) a then provided one can learn Q it is not necessary to have knowledge of S and R to obtain the optimal policy. As we are interested in the case where these functions are not known, we need something slightly different.

4 The Q function Note also that V opt (s) = max Q(s,) and so Q(s,a) = R(s, a) + ǫ max Q(S(s, a),) which suggests a simple learning algorithm. Let Q be our learner s estimate of what the exact Q function is. That is, in the current scenario Q is a table containing the estimated values of Q(s,a) for all pairs (s, a). Q-learning Start with all entries in Q set to 0. (In fact we will see in a moment that random entries will do.) Repeat the following: 1. Look at the current state s and choose an action a. (We will see how to do this in a moment.) 2. Do the action a and obtain some reward R(s,a). 3. Observe the new state S(s, a). 4. Perform the update Q (s, a) = R(s, a) + ǫ max Q (S(s, a),) Note that this can be done in episodes. For example, in learning to play games, we can play multiple games, each being a single episode. This looks as though it might converge! Note that, if the rewards are at least 0 and we initialise Q to 0 then, n,s, a Q n+1(s, a) Q n(s,a) and n,s, a Q(s, a) Q n(s, a) 0 However, we need to be a bit more rigorous than this... If: 1. the agent is operating in an environment that is a deterministic MDP; 2. rewards are bounded in the sense that there is a constant δ > 0 such that s,a R(s,a) < δ 3. all possible pairs s and a are visited infinitely often; then the Q-learning algorithm converges, in the sense that as n. a,s Q n(s, a) Q(s,a)

5 This is straightforward to demonstrate. Using condition 3, take two stretches of time in which all s and a pairs occur: All s, a occur All s, a occur Define ξ(n) = max s,a Q n(s, a) Q(s,a) the maximum error in Q at n. We have, Q n+1(s, a) Q(s,a) = (R(s,a) ǫ max = ǫ max Q n(s(s, a),) max Q n(s(s, a),) Q(S(s,a),) ǫ max ǫ max s,a Q n(s, a) Q(s,a) = ǫξ(n). Convergence as described follows. Q n(s(s, a),)) (R(s, a) ǫ max Q(S(s, a),)) Q(S(s,a),) What happens when Q n(s, a) is updated to Q n+1(s, a)? Choosing actions to perform We have not yet answered the question of how to choose actions to perform during learning. One approach is to choose actions based on our current estimate Q. For instance action chosen in current state s = argmaxq (s, a). a However we have already noted the trade-off between exploration and exploitation. It makes more sense to: explore during the early stages of training; exploit during the later stages of training. Choosing actions to perform One way in which to choose actions that incorporates these requirements is to introduce a constant λ and choose actions probabilistically according to Note that: Pr(action a state s) = if λ is small this promotes exploration; if λ is large this promotes exploitation. We can vary λ as training progresses. λq (s,a) a λq (s,a) This seems particularly important in the light of condition 3 of the convergence proof.

6 Improving the training process There are two simple ways in which the process can be improved: 1. If training is episodic, we can store the rewards obtained during an episode and update backwards at the end. This allows better updating at the expense of requiring more memory. 2. We can remember information about rewards and occasionally re-use it by re-training. Nondeterministic MDPs The Q-learning algorithm generalises easily to a more realistic situation, where the outcomes of actions are probabilistic. Instead of the functions S and R we have probability distributions and Pr(new state current state, action) Pr(reward current state, action). and we now use S(s, a) and R(s, a) to denote the corresponding random variables. We now have ( ) V p = E ǫ i r t+i and the best policy p opt maximises V p. We now have Q-learning for nondeterministic MDPs Q(s, a) = E(R(s,a)) + ǫ σ = E(R(s,a)) + ǫ σ Pr(σ s,a)v opt (σ) Pr(σ s, a) max Q(σ,) and the rule for learning becomes ] Q n+1 = (1 θ n+1 )Q n(s, a) + θ n+1 [R(s, a) + max Q n(s(s, a),) with θ n+1 = v n+1 (s, a) where v n+1 (s, a) is the number of times the pair s and a has been visited so far. If: for nondeterministic MDPs 1. the agent is operating in an environment that is a nondeterministic MDP; 2. rewards are bounded in the sense that there is a constant δ > 0 such that s,a R(s,a) < δ 3. all possible pairs s and a are visited infinitely often; 4. n i (s, a) is the ith time that we do action a in state s; and also...

7 ...we have for nondeterministic MDPs 0 θ n < 1 θ ni (s,a) = i=1 θn 2 i (s,a) < i=1 then with probability 1 the Q-learning algorithm converges, in the sense that a,s Q n(s, a) Q(s,a) as n. 26 Alternative representation for the Q table But there s always a catch... We have to store the table for Q : even for quite straightforward problems it is HUGE!!! - certainly big enough that it can t be stored; a standard approach to this problem is, for example, to represent it as a neural network; one way might be to make s and a the inputs to the network and train it to produce Q (s, a) as its output. This, of course, introduces its own problems, although it has been used very successfully in practice. It might be covered in Artificial Intelligence III, which unfortunately does not yet exist!